Easing the Monte Carlo sign problem

We introduce a procedure to systematically search for a local unitary transformation that maps a wave function with a nontrivial sign structure into a positive-real form. The transformation is parametrized as a quantum circuit compiled into a set of one-and two-qubit gates. We design a cost function that maximizes the average sign of the output state and removes its complex phases. The optimization of the gates is performed through automatic differentiation algorithms, widely used in the machine learning community. We provide numerical evidence for significant improvements in the average sign for a two-leg triangular Heisenberg ladder with next-to-nearestneighbor and ring-exchange interactions. This model exhibits phases where the sign structure can be removed by simple local one-qubit unitaries, but also an exotic Bose-metal phase whose sign structure induces "Bose surfaces" with a fermionic character and a higher entanglement that requires deeper circuits.

Section: Discussionmentioning

confidence: 99%

Wave-function positivization via automatic differentiation

Torlai

Carrasquilla

Fishman

et al. 2020

“…That is, despite the sign problem, it may still be possible to access large enough systems to extract the relevant physics. Moreover, there are several recent works focusing on easing the sign problem to bring a larger set of problems within the reach of current technology [68,69]. The application of machine learning techniques has also found success beyond QMC for systems with a fermionic sign problem [70,71].…”

Section: Discussionmentioning

confidence: 99%

Intrinsic sign problems in topological quantum field theories

Smith

Golan

Ringel

2020

The sign problem is a widespread numerical hurdle preventing us from simulating the equilibrium behavior of various problems at the forefront of physics. Focusing on an important subclass of such problems, bosonic (2 + 1)-dimensional topological quantum field theories, here we provide a simple criterion to diagnose intrinsic sign problems-that is, sign problems that are inherent to that phase of matter and cannot be removed by any local unitary transformation. Explicitly, if the exchange statistics of the anyonic excitations do not form complete sets of roots of unity, then the model has an intrinsic sign problem. This establishes a concrete connection between the statistics of anyons, contained in the modular S and T matrices, and the presence of a sign problem in a microscopic Hamiltonian. Furthermore, it places constraints on the phases that can be realized by stoquastic Hamiltonians. We prove this and a more restrictive criterion for the large set of gapped bosonic models described by an Abelian topological quantum field theory at low-energy, and we offer evidence that it applies more generally with analogous results for non-Abelian and chiral theories.

“…The severity of a sign problem is quantified by the smallness of the average sign of the QMC weights p with respect to the distribution |p|, i.e., sgn := p/ |p|. Since sgn can be viewed as the ratio of two partition functions, it obeys the generic scaling sgn ∼ e − βN , with 0, as βN → ∞ [2,10]. A sign problem exists when > 0, in which case QMC simulations require exponential computational resources, and this is what the intrinsic sign problem we identified implies for "most" chiral topological phases of matter.…”

Section: Discussion and Outlookmentioning

confidence: 99%

“…Conceptually, the sign problem can be understood as an obstruction to mapping quantum systems to classical systems, and accordingly, from a number of complexity theoretic perspectives, a generic curing algorithm in polynomial time is not believed to exist [2,[6][7][8][9][10]. In many-body physics, however, one is mostly interested in universal phenomena, i.e., phases of matter and the transitions between them, and therefore representative Hamiltonians which are sign-free often suffice [11].…”

Section: Introductionmentioning

confidence: 99%

“…Designing sign-free models requires design principles (or "de-sign" principles) [11,18]-easily verifiable properties that, if satisfied by a Hamiltonian and method, lead to a signfree representation of the corresponding partition function. An important example is the condition i|H| j 0 where i = j label a local basis, which implies non-negative weights p in a wide range of methods [10,11]. Hamiltonians satisfying this condition in a given basis are known as stoquastic [8], and have proven very useful in both application and theory of QMC in bosonic (or spin, or "qudit") systems [2,[6][7][8][9][10][11].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Intrinsic sign problem in fermionic and bosonic chiral topological matter

Golan

Smith

Ringel

2020

The infamous sign problem leads to an exponential complexity in Monte Carlo simulations of generic many-body quantum systems. Nevertheless, many phases of matter are known to admit a sign-problem-free representative, allowing efficient simulations on classical computers. Motivated by long-standing open problems in many-body physics, as well as fundamental questions in quantum complexity, the possibility of intrinsic sign problems, where a phase of matter admits no sign-problem-free representative, was recently raised but remains largely unexplored. Here we establish the existence of an intrinsic sign problem in a broad class of gapped, chiral, topological phases of matter. Within this class, we exclude the possibility of stoquastic Hamiltonians for bosons (or "qudits") and of sign-problem-free determinantal Monte Carlo algorithms for fermions. The intrinsically sign-problematic class of phases we identify is defined in terms of topological invariants with clear observable signatures: the chiral central charge and the topological spins of anyons. We obtain analogous results for phases that are spontaneously chiral, and present evidence for an extension of our results that applies to both chiral and nonchiral topological matter.